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Studying Null and Time-Like Geodesics in the Classroom Studying null and time-like geodesics in the classroom Thomas Muller¨ Visualisierungsinstitut der Universitat¨ Stuttgart (VISUS), Allmandring 19, 70569 Stuttgart, Germany E-mail: [email protected] Jorg¨ Frauendiener‡ Department of Mathematics & Statistics, University of Otago, P.O. Box 56, Dunedin 9010, New Zealand E-mail: [email protected] Abstract. In a first course of general relativity it is usually quite difficult for students to grasp the concept of a geodesic. It is supposed to be straight (auto-parallel) and yet it ‘looks’ curved. In these situations it is very useful to have some explicit examples available which show the different behaviour of geodesics. In this paper we present the GeodesicViewer, an interactive tool for studying the behaviour of geodesics in many different space-times. The geodesics can be represented in several ways, depending on the space-time in question. The use of a local reference frame and ‘Cartesian-like’ coordinates helps the students to develop some intuition in various situations. We present the various features of the GeodesicViewer in the form of readily formulated exercises for the students. PACS numbers: 04.20.-q Submitted to: Eur. J. Phys. 1. Introduction The intrinsic curvature of a space-time in general relativity is a concept that contradicts our every day experience of space and time. The necessary mathematical set of tools is difficult to learn and fairly abstract. To get some impression what a curved space-time means, is to study the behaviour of light rays and particles in free motion. In the geometric optics limit and for particles whose mass has no back-reaction on the curvature of space-time, light rays and particles in free motion can be represented by null and time-like geodesics, respectively. For a first glimpse on how null and time-like geodesics behave, it may be sufficient to arXiv:1105.0109v1 [physics.ed-ph] 30 Apr 2011 use off-the-shelf/standard software like for example Maple, Mathematica, or Octave. All of them could integrate the geodesic equation and show the geodesic as 2d- or 3d-plot. However, to explore the behaviour of geodesics, an interactive tool is indispensable. In this article we present the GeodesicViewer [1], an interactive tool to thoroughly examine the behaviour of light-like and time-like geodesics in a space-time whose metric ‡ and: Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, NO-0316 Oslo, Norway Studying null and time-like geodesics in the classroom 2 Figure 1. Screenshot of the GeodesicViewer’s user interface. is provided analytically. The database of metrics is taken from the Motion4D library [2]. The metrics with the corresponding Christoffel symbols and local tetrads are detailed in Ref. [3]. The graphical user interface, see Fig. 1, is written using the object-oriented, cross-platform application framework Qt[4]. The graphical 2D and 3D output is realized by means of the Open Graphics Library (OpenGL)[5]. For the numerical integration of the geodesics, we use a standard fourth-order Runge-Kutta method and the integrators of the GNU Scientific Library (GSL)[6]. The structure of the paper is as follows. In Sec. 2 we give a short description of the GeodesicViewer. Sec. 3 discusses the standard situations in the Schwarzschild space-time. Periodic orbits of time-like geodesics in black hole space-times are worked out in Sec. 4. Secs. 5 and 6 deal with the more exotic space-times like the Morris-Thorne wormhole and the extreme Reissner-Nordstrøm diblack hole. The GeodesicViewer is freely available for Linux and Windows. The source code and several examples can be downloaded from www.vis.uni-stuttgart.de/relativity. 2. GeodesicViewer The two main outputs of the GeodesicViewer are 3D and 2D representations of the geodesic data. In the standard 3D representation, a geodesic is depicted by means of pseudo-Cartesian coordinates, where the inherent coordinates are transformed into Cartesian coordinates as usual. If an embedding diagram is defined for a specific hypersurface of the space-time, the geodesic can also be represented in this form. (In principle, the user can implement any representation he wants.) The standard 2D representation of a geodesic is also given in pseudo-Cartesian coordinates, where now only a specific hypersurface is used. Another representation shows coordinate or velocity relations. For example, the radial coordinate could be plotted against the affine parameter. A third representation follows from the Euler- Lagrangian formalism, where an effective potential can be defined. This representation is particularly helpful to find bound orbits. The numerous features of the GeodesicViewer are described by means of examples which are formulated as exercises. The mathematical explanations are directed to the teacher. The Studying null and time-like geodesics in the classroom 3 exercises can be made either from scratch or a configure file can be prepared in advance, so that the student only has to change a few parameters. Each result description is accompanied by a configure file which holds the final result for reproduction in the GeodesicViewer. To familiarize oneself with the graphical user interface of the GeodesicViewer, there are several online tutorials. These can be worked through either alone or with guidance from the teacher. 3. Basic examples in the Schwarzschild space-time The prime example of general relativity is the Schwarzschild metric which we will give here m 2 m n in isotropic coordinates x = (t;x;y;z). The line element ds = gmn dx dx reads 2 4 1 − rs=r rs ds2 = − c2dt2 + 1 + dx2 + dy2 + dz2; (1) 1 + rs=r r 2 2 2 2 2 where r = x +y +z , rs = GM=(2c ) is the Schwarzschild radius, G is Newton’s constant, M is the mass of the black hole, and c is the speed of light. The transformation between the usual Schwarzschild radial coordinate r and the isotropic radial coordinate r is given 2 by r = r (1 + rs=r) . If M = 0, Eq. (1) simplifies to the Minkowski metric in Cartesian coordinates. Because of the spherical symmetry of the Schwarzschild space-time, we can restrict geodesics to the xy-plane. Then, the geodesic equations read t t 0 = t¨+ 2Gtxt˙x˙+ 2Gtyt˙y˙; (2) x 2 x 2 x x 2 0 = x¨+ Gttt˙ + Gxxx˙ + 2Gxyx˙y˙+ Gyyy˙ ; (3) y 2 y 2 y y 2 0 = y¨+ Gttt˙ + Gxxx˙ + 2Gxyx˙y˙+ Gyyy˙ ; (4) with the Christoffel symbols 2 3 2 3 x 2c r rs (r − rs)x y 2c r rs (r − rs)y Gtt = 7 ; Gtt = 7 ; (5) (r + rs) (r + rs) t 2rsx t 2rsy Gtx = 3 2 2 ; Gty = 3 2 2 ; (6) r [1 − rs =r ] r [1 − rs =r ] x y x 2rs x Gxx = Gxy = −Gyy = − 3 ; (7) r 1 + rs=r y y x 2rs y Gyy = −Gxx = Gxy = − 3 : (8) r 1 + rs=r Here, a dot represents the derivative with respect to the affine parameter l, hence t˙ = dt=dl. To integrate the geodesic equations, we need not only an initial position but also an initial direction. For this purpose, we first introduce the local reference frame fe(i)gi=t;x;y;z of m an observer which defines a local Minkowskian system. The four base vectors e(i) = e(i)¶m m n have to fulfill the orthonormality condition gmn e(i)e( j) = h(i)( j) with h(i)( j) = diag(−1;1;1;1). Here, the most convenient choice for the local reference frame is the one which is adapted to the coordinates and the symmetries of the metric, −2 1 + rs=r ¶t rs e(t) = ; e(x) = 1 + ¶x; (9) 1 − rs=r c r −2 −2 rs rs e = 1 + ¶y; e = 1 + ¶z: (10) (y) r (z) r Studying null and time-like geodesics in the classroom 4 e(3) k χ ξ e(2) e(1) (i) Figure 2. Initial direction k = k e(i) with respect to the local reference frame of an observer. (i) m m Now, an initial direction, k = k e(i)¶m = k ¶m , of a null or a time-like geodesic can be defined with respect to this local reference frame as shown in Fig. 2. For a null geodesic we have k = e(0) + sin c cosxe(1) + sin c sinxe(2) + cos ce(3). The initial direction of a time-like geodesic, on the other hand, equals the initial four-velocity k = u = gce(0) +gbc sin c cosxe(1) + sin c sinxe(2) + cos ce(3) , where b = v=c is the initial p velocity v scaled by the speed of light, and g = 1= 1 − b 2. Because we restrict to geodesics in the xy-plane, we must set c = 90◦. The geodesic equations can now be solved using the xm xm km initial position l=0 and the initial direction ˙ l=0 = . The numerical integration of the geodesics are accomplished by either a standard fourth- order Runge-Kutta method or by the integrators of the GNU Scientific Library. After each m n 2 integration step, the geodesic is tested if it still fulfills the constraint equation gmn x˙ x˙ = kc with k = 0 for light-like and k = −1 for time-like geodesics. If this constraint is not fulfilled within a certain accuracy, the integration stops. For the following examples, we set G = c = M = 1. In physical units, this means that 2 5 G=c = 1 ls=M which equals one light second per mass unit M ≈ 2:03 × 10 M . Hence, M = 1 represents an object of 2:03 × 105 solar masses and distances are measured in light seconds.
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